Question 2 a. State the Refinement \( Q \) of a partition \( P \) b. Explain the fundamental theorem of calculus. c. Prove that a bounded function \( f:[a, b] \rightarrow \mathbb{R} \) is Riemann integrable if and only if for every \( \varepsilon>0 \) there exists a partition \( P \) of \( [a, b] \), which may depend on \( E \), such that \( U(f ; P)-L(f ; P)

Let's address each part of your question step by step. ### a. State the Refinement \( Q \) of a partition \( P \) A **refinement** \( Q \) of a partition \( P \) of an interval \([a, b]\) is a partition that includes all the points of \( P \) and possibly additional points. Formally, if \( P = \{x_0, x_1, \ldots, x_n\} \) is a partition of \([a, b]\), then a refinement \( Q \) is a partition such that \( P \subseteq Q \). This means that every point in \( P \) is also in \( Q\), and \( Q \) may contain more points, i.e., \( Q = \{x_0, x_1, \ldots, x_n, y_1, y_2, \ldots, y_m\} \) where \( y_i \) are additional points in \([a, b]\). ### b. Explain the Fundamental Theorem of Calculus The **Fundamental Theorem of Calculus** connects differentiation and integration, and it consists of two main parts: 1. **First Part**: If \( f \) is a continuous function on the interval \([a, b]\), then the function \( F \) defined by \[ F(x) = \int_{a}^{x} f(t) \, dt \] for \( x \in [a, b] \) is continuous on \([a, b]\) and differentiable on \( (a, b) \), and its derivative is given by \[ F'(x) = f(x). \] 2. **Second Part**: If \( f \) is integrable on \([a, b]\) and \( F \) is an antiderivative of \( f \) (i.e., \( F' = f \)), then \[ \int_{a}^{b} f(t) \, dt = F(b) - F(a). \] This theorem establishes that integration and differentiation are inverse processes. ### c. Prove that a bounded function \( f:[a, b] \rightarrow \mathbb{R} \) is Riemann integrable if and only if for every \( \varepsilon>0 \) there exists a partition \( P \) of \( [a, b] \) such that \( U(f ; P) - L(f ; P) < \varepsilon \). **Proof**: 1. **(⇒)** Assume \( f \) is Riemann integrable. By the definition of Riemann integrability, for every \( \varepsilon > 0 \), there exists a partition \( P \) such that the difference between the upper sum \( U(f; P) \) and the lower sum \( L(f; P) \) is less than \( \varepsilon \): \[ U(f; P) - L(f; P) < \varepsilon. \] 2. **(⇐)** Conversely, assume that for every \( \varepsilon > 0 \), there exists a partition \( P \) such that \( U(f; P) - L(f; P) < \varepsilon \). We need to show that \( f \) is Riemann integrable. By the definition of Riemann integrability, a bounded function is Riemann integrable if the set of discontinuities has measure zero. The condition \( U(f; P) - L(f; P) < \varepsilon \) implies that the oscillation of \( f \) over the intervals defined by the partition can be made arbitrarily small. This means that the discontinuities of \( f \) must be limited in a way that they do not contribute significantly to the upper and lower sums, which implies that the set of discontinuities must have measure zero. Therefore, \( f \) is Riemann integrable. Thus, we conclude that a bounded function \( f \) is Riemann integrable if and only if for every \( \varepsilon > 0 \), there exists a partition \( P \) such that \( U(f; P) - L(f; P) < \varepsilon \). ### d. Use the Fundamental Theorem of Calculus to compute the derivative of \( F(x) = \left(\int_{0}^{x} \cos(t) \, dt\right)^{3} \). To find the derivative \( F'(x) \), we can use the chain rule and the Fundamental Theorem of Calculus. 1. Let \( G(x) = \int_{0}^{x} \cos(t